Chapter 8.3, Problem 74E

A secant reduction formula Prove that for positive integers n ≠ 1,

${\displaystyle \int {\sec }^{n}xdx=\frac{{\sec }^{n-2}x\tan x}{n-1}+\frac{n-2}{n-1}{\displaystyle \int {\sec }^{n-2}xdx.}}$

(Hint: Integrate by parts with u = sec^{n − 2} x and dv = sec² x dx.)